It is generally believed that in spatial dimension \(d\) > 1 the leading contribution to the entanglement entropy \(S = - tr\rho_A log \rho_A\) scales as the area of the boundary of subsystem \(A\). The coefficient of this "area law" is non-universal. However, in the neighbourhood of a quantum critical point \(S\) is believed to possess subleading universal corrections. In the present work, we study the entanglement entropy in the quantum \(O(N)\) model in 1 < \(d\) < 3. We use an expansion in \(\epsilon = 3-d\) to evaluate i) the universal geometric correction to \(S\) for an infinite cylinder divided along a circular boundary; ii) the universal correction to \(S\) due to a finite correlation length. Both corrections are different at the Wilson-Fisher and Gaussian fixed points, and the \(\epsilon \to 0\) limit of the Wilson-Fisher fixed point is distinct from the Gaussian fixed point. In addition, we compute the correlation length correction to the Renyi entropy \(S_n = 1/1-n log tr {\rho_A}^n\) in \(\epsilon\) and large-\(N\) expansions. For \(N \to \infty\), this correction generally scales as \(N^2\) rather than the naively expected \(N\). Moreover, the Renyi entropy has a phase transition as a function of \(n\) for \(d\) close to 3.